Abstract
We obtain the Courant bracket twisted simultaneously by a 2-form B and a bi-vector theta by calculating the Poisson bracket algebra of the symmetry generator in the basis obtained acting with the relevant twisting matrix. It is the extension of the Courant bracket that contains well known Schouten–Nijenhuis and Koszul bracket, as well as some new star brackets. We give interpretation to the star brackets as projections on isotropic subspaces.
Highlights
The Courant bracket [1,2] represents the generalization of the Lie bracket on spaces of generalized vectors, understood as the direct sum of the elements of the tangent bundle and the elements of the cotangent bundle
We obtain the Courant bracket twisted simultaneously by a 2-form B and a bi-vector θ by calculating the Poisson bracket algebra of the symmetry generator in the basis obtained acting with the relevant twisting matrix
We examined various twists of the Courant bracket, that appear in the Poisson bracket algebra of symmetry generators written in a suitable basis, obtained acting on the double canonical variable (2.4) by the appropriate elements of O(D, D) group
Summary
The Courant bracket [1,2] represents the generalization of the Lie bracket on spaces of generalized vectors, understood as the direct sum of the elements of the tangent bundle and the elements of the cotangent bundle. We represent the symmetry generator in the basis obtained acting with the twisting matrix eBon the double canonical variable This generator is manifestly self T-dual and its algebra closes on the Courant bracket twisted by both B and θ. We proceed with rewriting it in the coordinate free notation, where many terms are recognized as the well known brackets, such as the Koszul or Schouten–Nijenhuis bracket, but some new brackets, that we call star brackets, appear These star brackets as a domain take the direct sum of tangent and cotangent bundle, and as a result give the graph of the bi-vector θin the cotangent bundle, i.e. the sub-bundle for which the vector and 1-form components are related as ξ μ = κθμνλν. The Courant bracket twisted by both B and θ and the one twisted by Care directly related by a O(D, D) transformation represented with the block diagonal matrix
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